Well, substitute for . You will have a function in terms of . So do that substitution and post what the result is.
Using the method of u-substitution,
u = 3x-7 (enter a function of x)
du = 3 dx (enter a function of x)
a = 2 (enter a number)
b = 5 (enter a number)
f(u) = __________(enter a function of u).
The value of the original integral is____________
so how would i know f(u) and please show steps and explain. so i can understand
What I was asking you to do is replace with where ever you see it in the equation.
Now, you can almost see , but you need to complete the substitution. If you look at the integral there is still a and we need a . So using the substitution
we see that
Now, make that substitution and reevaluate the limits, i.e. and
and finally the integral is , so
Also, the crucial thing to try and understand is that finding an anti-derivative (which is virtually your integral) involves working backwards through the chain rule for differentiation. So look at a similar case, a power of a polynomial, but consider differentiation first. Just in case a picture helps...
Can you differentiate with the chain rule? There's a formula with u, of course, but it amounts to filling out the blanks in this picture, where straight continuous lines differentiate downwards (integrate up) with respect to x, and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).
To find the anti-derivative (I) for your problem, we should start at the bottom...
... but, bearing in mind the effect of the chain rule coming 'down' (i.e. differentiating), we'll need to cancel the multiplying by 3, in order to have the lower equals sign...
Now get an anti-derivative with respect to the dashed balloon...
At this stage always differentiate your anti-derivative to see if you get your original expression (the 'integrand'). But of course with the picture you can just examine it downwards instead of up.
And then consider the definite integral...
Don't integrate - balloontegrate!
Balloon Calculus; standard integrals, derivatives and methods
Balloon Calculus Drawing with LaTeX and Asymptote!
You will learn more techniques to integrate. U-substitution is (probably) the simplest, so make sure you understand it!
I red that integrating is harder than differentiating because it is not apparent which technique you need to use. For me I always try to start with the simplest, that being u-substitution.